A221362 Number of distinct groups of order n that are the torsion subgroup of an elliptic curve over the rationals Q.

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

1,4

Comments

Barry Mazur proved that the torsion subgroup of an elliptic curve over Q is one of the 15 following groups: Z/NZ for N = 1, 2, …, 10, or 12, or Z/2Z × Z/2NZ with N = 1, 2, 3, 4.

References

J. H. Silverman, The Arithmetic of Elliptic Curves, Graduates Texts in Mathematics 106, Springer-Verlag, 1986 (see Theorem 7.5).

Links

Table of n, a(n) for n=1..37.

B. Mazur, Rational isogenies of prime degree, Inventiones Math. 44, 2 (June 1978), 129-162.

Wikipedia, Elliptic curve

Wikipedia, Mazur's torsion theorem

Formula

a(n) = 0 for n > 16.

a(A059765(n)) > 0. - Jonathan Sondow, May 10 2014

Example

a(4) = 2 because a subgroup of order 4 in an elliptic curve over Q is isomorphic to one of the 2 groups Z/4Z or Z/2Z × Z/2Z.

Crossrefs

Cf. A059765 (possible sizes of the torsion subgroup of an elliptic curve over Q), A146879.

Sequence in context: A327785 A105242 A336709 * A114116 A054532 A260415

Adjacent sequences: A221359 A221360 A221361 * A221363 A221364 A221365

Keyword

nonn,fini,full,easy

Author

Jonathan Sondow, Jan 12 2013

Status

approved